Proving Lines Parallel

Warm UpState the converse of each statement.

1. If a = b, then a + c = b + c.

2. If mA + mB = 90°, then A and B are complementary.

3. If AB + BC = AC, then A, B, and C are collinear.

If a + c = b + c, then a = b.

If A and B are complementary, then mA + mB =90°.

If A, B, and C are collinear, then AB + BC = AC.

Proving Lines ParallelSame Side Interior

• With a transversal passing through parallel lines, two same side angles will add up to 180

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Use the angles formed by a transversal to prove two lines are parallel.

Objective

Proving Lines Parallel

Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem.

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

Example 1A: Using the Converse of the Corresponding Angles Postulate

4 8

4 8 4 and 8 are corresponding angles. ℓ || m Conv. of Corr. s Post.

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Proving Lines Parallel

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

Example 1B: Using the Converse of the Corresponding Angles Postulate

m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30

m3 = 4(30) – 80 = 40 Substitute 30 for x.m8 = 3(30) – 50 = 40 Substitute 30 for x.

ℓ || m Conv. of Corr. s Post.3 8 Def. of s.

m3 = m8 Trans. Prop. of Equality

Proving Lines Parallel

Proving Lines ParallelCheck It Out! Example 1a

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

m1 = m3

1 3 1 and 3 are corresponding angles. ℓ || m Conv. of Corr. s Post.

Proving Lines ParallelCheck It Out! Example 1b

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

m7 = (4x + 25)°,

m5 = (5x + 12)°, x = 13

m7 = 4(13) + 25 = 77 Substitute 13 for x.m5 = 5(13) + 12 = 77 Substitute 13 for x.

ℓ || m Conv. of Corr. s Post.7 5 Def. of s.

m7 = m5 Trans. Prop. of Equality

Proving Lines Parallel

The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.

Proving Lines Parallel

Use the given information and the theorems you have learned to show that r || s.

Example 2A: Determining Whether Lines are Parallel

4 8

4 8 4 and 8 are alternate exterior angles.

r || s Conv. Of Alt. Int. s Thm.

Proving Lines Parallel

m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5

Use the given information and the theorems you have learned to show that r || s.

Example 2B: Determining Whether Lines are Parallel

m2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x.

m3 = 25x – 3 = 25(5) – 3 = 122 Substitute 5 for x.

Proving Lines Parallel

m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5

Use the given information and the theorems you have learned to show that r || s.

Example 2B Continued

r || s Conv. of Same-Side Int. s Thm.

m2 + m3 = 58° + 122°= 180° 2 and 3 are same-side

interior angles.

Proving Lines ParallelCheck It Out! Example 2a

m4 = m8

Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.

4 8 4 and 8 are alternate exterior angles.

r || s Conv. of Alt. Int. s Thm.

4 8 Congruent angles

Proving Lines ParallelCheck It Out! Example 2b

Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.

m3 = 2x, m7 = (x + 50), x = 50

m3 = 100 and m7 = 1003 7 r||s Conv. of the Alt. Int. s Thm.

m3 = 2x = 2(50) = 100° Substitute 50 for x.m7 = x + 50 = 50 + 50 = 100° Substitute 5 for x.

Proving Lines ParallelExample 3: Proving Lines Parallel

Given: p || r , 1 3Prove: ℓ || m

Proving Lines ParallelExample 3 Continued

Statements Reasons

1. p || r

5. ℓ ||m

2. 3 23. 1 34. 1 2

2. Alt. Ext. s Thm.1. Given

3. Given4. Trans. Prop. of 5. Conv. of Corr. s Post.

Proving Lines ParallelCheck It Out! Example 3

Given: 1 4, 3 and 4 are supplementary.Prove: ℓ || m

Proving Lines ParallelCheck It Out! Example 3 Continued

Statements Reasons

1. 1 4 1. Given2. m1 = m4 2. Def. s 3. 3 and 4 are supp. 3. Given4. m3 + m4 = 180 4. Trans. Prop. of 5. m3 + m1 = 180 5. Substitution6. m2 = m3 6. Vert.s Thm.7. m2 + m1 = 180 7. Substitution8. ℓ || m 8. Conv. of Same-Side

Interior s Post.

Proving Lines ParallelExample 4: Carpentry Application

A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.

Proving Lines ParallelExample 4 Continued

A line through the center of the horizontal piece forms a transversal to pieces A and B.

1 and 2 are same-side interior angles. If 1 and 2 are supplementary, then pieces A and B are parallel.

Substitute 15 for x in each expression.

Proving Lines ParallelExample 4 Continued

m1 = 8x + 20= 8(15) + 20 = 140

m2 = 2x + 10= 2(15) + 10 = 40

m1+m2 = 140 + 40= 180

Substitute 15 for x.

Substitute 15 for x.

1 and 2 are supplementary.

The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.

Proving Lines ParallelCheck It Out! Example 4

What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel.

4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30°The angles are congruent, so the oars are || by the Conv. of the Corr. s Post.

Proving Lines ParallelLesson Quiz: Part I

Name the postulate or theoremthat proves p || r.

1. 4 5 Conv. of Alt. Int. s Thm.

2. 2 7 Conv. of Alt. Ext. s Thm.

3. 3 7 Conv. of Corr. s Post.

4. 3 and 5 are supplementary.

Conv. of Same-Side Int. s Thm.

Proving Lines ParallelLesson Quiz: Part II

Use the theorems and given information to prove p || r.

5. m2 = (5x + 20)°, m 7 = (7x + 8)°, and x = 6m2 = 5(6) + 20 = 50°m7 = 7(6) + 8 = 50°m2 = m7, so 2 ≅ 7

p || r by the Conv. of Alt. Ext. s Thm.